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I am reading "Classical Mechanics" by John Taylor.

There is the following problem in this book:

In Section 1.5 we proved that Newton's third law implies the conservation of momentum. Prove the converse, that if the law of conservation of momentum applies to every possible group of particles, then the interparticle forces must obey the third law. [Hint: However many particles your system contains, you can focus your attention on just two of them. (Call them $1$ and $2$.) The law of conservation of momentum says that if there are no external forces on this pair of particles, then their total momentum must be constant. Use this to prove that $F_{12} = -F_{21}$.]

It is true that $F_{12} = -F_{21}$ holds when there are no external forces on this pair of particles.

Then, can we say the following statement?

$F_{12} = -F_{21}$ holds when there exist external forces on this pair of particles.

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By notation external forces are not included, so $F_{21} + F_{12} = 0$ always holds, even if there are external forces. It is $F_{1} + F_{2}$ that is not zero if there is an external force that is not opposite and equal on both particles. Note that the magnetic part of the Lorentz "force" does not obey the third law and does not conserve momentum.

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  • $\begingroup$ I think OP's concern is that although we can show $\mathbf{F}_{12}+\mathbf{F}_{21}=0$ when the net external force on both the particles is zero by conservation of momentum, what is the guarantee that the internal forces will still be equal and opposite in the presence of a net external force? What if the internal forces change their behavior so that $\mathbf{F}_{12} \neq -\mathbf{F}_{21}$ when a net external force on the two-particle system is present? (I know that it sounds absolutely absurd to even encourage this skepticism, but we need cover all possibilities when proving something) $\endgroup$ – Ajay Mohan Nov 3 '19 at 9:48
  • $\begingroup$ @AjayMohan As I said, $F_{21} + F_{12} = 0$ always holds, external forces or not. $\endgroup$ – my2cts Nov 3 '19 at 9:54
  • $\begingroup$ Can you show that from merely this statement "Total momentum of two particles is conserved when there is no net external force acting on the two-particle system"? $\endgroup$ – Ajay Mohan Nov 3 '19 at 9:57
  • $\begingroup$ @AjayMohan Clearly formulated. No you can't. If there were a force $F_{123}$ that can not be broken down into two particle forces then the statement might be considered wrong. Physics has not encountered this as far as I know. The textbook does not do a good job here. $\endgroup$ – my2cts Nov 3 '19 at 10:01

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